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\date{Due: November 12, 2025}

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\noindent \textbf{Homework Assignment 3\hfill Math 369 - Introduction to Abstract Algebra}
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\noindent\textit{``Rigour is to the mathematician what morality is to men.''}   \hfill --- Andr\'e Weil
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\noindent \textbf{Instructions:} Show all work clearly. Justify your answers with complete explanations. Late assignments will not be accepted without prior arrangement.

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\noindent \textbf{Problem 1.} Let $G$ be an abelian group. Define
$$H = \{x^2 : x \in G\} \quad \text{and} \quad K = \{x \in G : x^2 = e\}.$$
Prove that $H \cong G/K$.

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\noindent \textbf{Problem 2.} Let $G$ and $H$ be groups. Suppose $J$ is a normal subgroup of $G$ and $K$ is a normal subgroup of $H$. Prove that
$$(G \times H)/(J \times K) \cong (G/J) \times (H/K).$$

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\noindent \textbf{Problem 3.} Consider the factor group $\mathbb{R}/\mathbb{Z}$ where $\mathbb{R}$ is the additive group of real numbers and $\mathbb{Z}$ is the subgroup of integers.

\begin{enumerate}
\item[(a)] Show that $\mathbb{Z}$ is a normal subgroup of $\mathbb{R}$.
\item[(b)] Prove that every coset $r + \mathbb{Z}$ has a unique representative in the interval $[0, 1)$.
\item[(c)] Consider the unit circle group $C = \{$ angles in $[0, 2\pi)\}$ under addition modulo $2\pi$ (where two angles are equivalent if they differ by a multiple of $2\pi$). Find an isomorphism $\phi: \mathbb{R}/\mathbb{Z} \to C$ and prove that it is indeed an isomorphism.
\item[(d)] Find all elements of order 2 or 3 in $\mathbb{R}/\mathbb{Z}$.
\item[(e)] Show that $[\pi] = \pi + \mathbb{Z}$ has infinite order in $\mathbb{R}/\mathbb{Z}$.
\end{enumerate}

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\noindent \textbf{Problem 4.} Let $p(x) = x^2 - 3 \in \mathbb{Q}[x]$ and let $E = \mathbb{Q}[x]/\langle x^2 - 3 \rangle$.

\begin{enumerate}
\item[(a)] Show that $p(x)$ is irreducible over $\mathbb{Q}$ and conclude that $E$ is a field.
\item[(b)] Compute the following products in $E$, expressing your answers in the form $[a + bx]$ where $a, b \in \mathbb{Q}$:
\begin{enumerate}
\item[(i)] $[2 + x] \cdot [1 - x]$
\item[(ii)] $[1 + x]^2$
\item[(iii)] $[x]^3$
\end{enumerate}
\item[(c)] Find the multiplicative inverse of $[2 + x]$ in $E$.
\item[(d)] Show that $E \cong \mathbb{Q}(\sqrt{3})$ by describing an explicit isomorphism.
\end{enumerate}

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\noindent \textbf{Problem 5.} Consider $p(x) = x^2 + x + 2 \in \mathbb{Z}_3[x]$ and let $F = \mathbb{Z}_3[x]/\langle x^2 + x + 2 \rangle$.

\begin{enumerate}
\item[(a)] Verify that $p(x)$ is irreducible over $\mathbb{Z}_3$ and determine $|F|$.
\item[(b)] List all elements of $F$ in the form $[a + bx]$ where $a, b \in \mathbb{Z}_3$.
\item[(c)] Compute $[x]^3$, $[1+x]^2$, and $[2+x] \cdot [1+2x]$ in $F$.
\item[(d)] Show that $F^\times$ is cyclic by finding a generator, where $F^\times = F \setminus \{[0]\}$ is the multiplicative group of nonzero elements.
\end{enumerate}

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\noindent \textbf{Problem 6.} Use the Euclidean algorithm to find the greatest common divisor of
$$f(x) = x^7 + 3x^5 + 8x^3 + x^2 + 24x + 3 \quad \text{and} \quad g(x) = x^4 + x^3 + 4x^2 + 3x + 3$$
in $\mathbb{Q}[x]$. Show all steps of your computation.

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